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<FONT color="green">001</FONT>    /*<a name="line.1"></a>
<FONT color="green">002</FONT>     * Licensed to the Apache Software Foundation (ASF) under one or more<a name="line.2"></a>
<FONT color="green">003</FONT>     * contributor license agreements.  See the NOTICE file distributed with<a name="line.3"></a>
<FONT color="green">004</FONT>     * this work for additional information regarding copyright ownership.<a name="line.4"></a>
<FONT color="green">005</FONT>     * The ASF licenses this file to You under the Apache License, Version 2.0<a name="line.5"></a>
<FONT color="green">006</FONT>     * (the "License"); you may not use this file except in compliance with<a name="line.6"></a>
<FONT color="green">007</FONT>     * the License.  You may obtain a copy of the License at<a name="line.7"></a>
<FONT color="green">008</FONT>     *<a name="line.8"></a>
<FONT color="green">009</FONT>     *      http://www.apache.org/licenses/LICENSE-2.0<a name="line.9"></a>
<FONT color="green">010</FONT>     *<a name="line.10"></a>
<FONT color="green">011</FONT>     * Unless required by applicable law or agreed to in writing, software<a name="line.11"></a>
<FONT color="green">012</FONT>     * distributed under the License is distributed on an "AS IS" BASIS,<a name="line.12"></a>
<FONT color="green">013</FONT>     * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.<a name="line.13"></a>
<FONT color="green">014</FONT>     * See the License for the specific language governing permissions and<a name="line.14"></a>
<FONT color="green">015</FONT>     * limitations under the License.<a name="line.15"></a>
<FONT color="green">016</FONT>     */<a name="line.16"></a>
<FONT color="green">017</FONT>    package org.apache.commons.math3.analysis.solvers;<a name="line.17"></a>
<FONT color="green">018</FONT>    <a name="line.18"></a>
<FONT color="green">019</FONT>    import org.apache.commons.math3.util.FastMath;<a name="line.19"></a>
<FONT color="green">020</FONT>    import org.apache.commons.math3.exception.NumberIsTooLargeException;<a name="line.20"></a>
<FONT color="green">021</FONT>    import org.apache.commons.math3.exception.NoBracketingException;<a name="line.21"></a>
<FONT color="green">022</FONT>    import org.apache.commons.math3.exception.TooManyEvaluationsException;<a name="line.22"></a>
<FONT color="green">023</FONT>    <a name="line.23"></a>
<FONT color="green">024</FONT>    /**<a name="line.24"></a>
<FONT color="green">025</FONT>     * This class implements the &lt;a href="http://mathworld.wolfram.com/MullersMethod.html"&gt;<a name="line.25"></a>
<FONT color="green">026</FONT>     * Muller's Method&lt;/a&gt; for root finding of real univariate functions. For<a name="line.26"></a>
<FONT color="green">027</FONT>     * reference, see &lt;b&gt;Elementary Numerical Analysis&lt;/b&gt;, ISBN 0070124477,<a name="line.27"></a>
<FONT color="green">028</FONT>     * chapter 3.<a name="line.28"></a>
<FONT color="green">029</FONT>     * &lt;p&gt;<a name="line.29"></a>
<FONT color="green">030</FONT>     * Muller's method applies to both real and complex functions, but here we<a name="line.30"></a>
<FONT color="green">031</FONT>     * restrict ourselves to real functions.<a name="line.31"></a>
<FONT color="green">032</FONT>     * This class differs from {@link MullerSolver} in the way it avoids complex<a name="line.32"></a>
<FONT color="green">033</FONT>     * operations.&lt;/p&gt;<a name="line.33"></a>
<FONT color="green">034</FONT>     * Muller's original method would have function evaluation at complex point.<a name="line.34"></a>
<FONT color="green">035</FONT>     * Since our f(x) is real, we have to find ways to avoid that. Bracketing<a name="line.35"></a>
<FONT color="green">036</FONT>     * condition is one way to go: by requiring bracketing in every iteration,<a name="line.36"></a>
<FONT color="green">037</FONT>     * the newly computed approximation is guaranteed to be real.&lt;/p&gt;<a name="line.37"></a>
<FONT color="green">038</FONT>     * &lt;p&gt;<a name="line.38"></a>
<FONT color="green">039</FONT>     * Normally Muller's method converges quadratically in the vicinity of a<a name="line.39"></a>
<FONT color="green">040</FONT>     * zero, however it may be very slow in regions far away from zeros. For<a name="line.40"></a>
<FONT color="green">041</FONT>     * example, f(x) = exp(x) - 1, min = -50, max = 100. In such case we use<a name="line.41"></a>
<FONT color="green">042</FONT>     * bisection as a safety backup if it performs very poorly.&lt;/p&gt;<a name="line.42"></a>
<FONT color="green">043</FONT>     * &lt;p&gt;<a name="line.43"></a>
<FONT color="green">044</FONT>     * The formulas here use divided differences directly.&lt;/p&gt;<a name="line.44"></a>
<FONT color="green">045</FONT>     *<a name="line.45"></a>
<FONT color="green">046</FONT>     * @version $Id: MullerSolver.java 1391927 2012-09-30 00:03:30Z erans $<a name="line.46"></a>
<FONT color="green">047</FONT>     * @since 1.2<a name="line.47"></a>
<FONT color="green">048</FONT>     * @see MullerSolver2<a name="line.48"></a>
<FONT color="green">049</FONT>     */<a name="line.49"></a>
<FONT color="green">050</FONT>    public class MullerSolver extends AbstractUnivariateSolver {<a name="line.50"></a>
<FONT color="green">051</FONT>    <a name="line.51"></a>
<FONT color="green">052</FONT>        /** Default absolute accuracy. */<a name="line.52"></a>
<FONT color="green">053</FONT>        private static final double DEFAULT_ABSOLUTE_ACCURACY = 1e-6;<a name="line.53"></a>
<FONT color="green">054</FONT>    <a name="line.54"></a>
<FONT color="green">055</FONT>        /**<a name="line.55"></a>
<FONT color="green">056</FONT>         * Construct a solver with default accuracy (1e-6).<a name="line.56"></a>
<FONT color="green">057</FONT>         */<a name="line.57"></a>
<FONT color="green">058</FONT>        public MullerSolver() {<a name="line.58"></a>
<FONT color="green">059</FONT>            this(DEFAULT_ABSOLUTE_ACCURACY);<a name="line.59"></a>
<FONT color="green">060</FONT>        }<a name="line.60"></a>
<FONT color="green">061</FONT>        /**<a name="line.61"></a>
<FONT color="green">062</FONT>         * Construct a solver.<a name="line.62"></a>
<FONT color="green">063</FONT>         *<a name="line.63"></a>
<FONT color="green">064</FONT>         * @param absoluteAccuracy Absolute accuracy.<a name="line.64"></a>
<FONT color="green">065</FONT>         */<a name="line.65"></a>
<FONT color="green">066</FONT>        public MullerSolver(double absoluteAccuracy) {<a name="line.66"></a>
<FONT color="green">067</FONT>            super(absoluteAccuracy);<a name="line.67"></a>
<FONT color="green">068</FONT>        }<a name="line.68"></a>
<FONT color="green">069</FONT>        /**<a name="line.69"></a>
<FONT color="green">070</FONT>         * Construct a solver.<a name="line.70"></a>
<FONT color="green">071</FONT>         *<a name="line.71"></a>
<FONT color="green">072</FONT>         * @param relativeAccuracy Relative accuracy.<a name="line.72"></a>
<FONT color="green">073</FONT>         * @param absoluteAccuracy Absolute accuracy.<a name="line.73"></a>
<FONT color="green">074</FONT>         */<a name="line.74"></a>
<FONT color="green">075</FONT>        public MullerSolver(double relativeAccuracy,<a name="line.75"></a>
<FONT color="green">076</FONT>                            double absoluteAccuracy) {<a name="line.76"></a>
<FONT color="green">077</FONT>            super(relativeAccuracy, absoluteAccuracy);<a name="line.77"></a>
<FONT color="green">078</FONT>        }<a name="line.78"></a>
<FONT color="green">079</FONT>    <a name="line.79"></a>
<FONT color="green">080</FONT>        /**<a name="line.80"></a>
<FONT color="green">081</FONT>         * {@inheritDoc}<a name="line.81"></a>
<FONT color="green">082</FONT>         */<a name="line.82"></a>
<FONT color="green">083</FONT>        @Override<a name="line.83"></a>
<FONT color="green">084</FONT>        protected double doSolve()<a name="line.84"></a>
<FONT color="green">085</FONT>            throws TooManyEvaluationsException,<a name="line.85"></a>
<FONT color="green">086</FONT>                   NumberIsTooLargeException,<a name="line.86"></a>
<FONT color="green">087</FONT>                   NoBracketingException {<a name="line.87"></a>
<FONT color="green">088</FONT>            final double min = getMin();<a name="line.88"></a>
<FONT color="green">089</FONT>            final double max = getMax();<a name="line.89"></a>
<FONT color="green">090</FONT>            final double initial = getStartValue();<a name="line.90"></a>
<FONT color="green">091</FONT>    <a name="line.91"></a>
<FONT color="green">092</FONT>            final double functionValueAccuracy = getFunctionValueAccuracy();<a name="line.92"></a>
<FONT color="green">093</FONT>    <a name="line.93"></a>
<FONT color="green">094</FONT>            verifySequence(min, initial, max);<a name="line.94"></a>
<FONT color="green">095</FONT>    <a name="line.95"></a>
<FONT color="green">096</FONT>            // check for zeros before verifying bracketing<a name="line.96"></a>
<FONT color="green">097</FONT>            final double fMin = computeObjectiveValue(min);<a name="line.97"></a>
<FONT color="green">098</FONT>            if (FastMath.abs(fMin) &lt; functionValueAccuracy) {<a name="line.98"></a>
<FONT color="green">099</FONT>                return min;<a name="line.99"></a>
<FONT color="green">100</FONT>            }<a name="line.100"></a>
<FONT color="green">101</FONT>            final double fMax = computeObjectiveValue(max);<a name="line.101"></a>
<FONT color="green">102</FONT>            if (FastMath.abs(fMax) &lt; functionValueAccuracy) {<a name="line.102"></a>
<FONT color="green">103</FONT>                return max;<a name="line.103"></a>
<FONT color="green">104</FONT>            }<a name="line.104"></a>
<FONT color="green">105</FONT>            final double fInitial = computeObjectiveValue(initial);<a name="line.105"></a>
<FONT color="green">106</FONT>            if (FastMath.abs(fInitial) &lt;  functionValueAccuracy) {<a name="line.106"></a>
<FONT color="green">107</FONT>                return initial;<a name="line.107"></a>
<FONT color="green">108</FONT>            }<a name="line.108"></a>
<FONT color="green">109</FONT>    <a name="line.109"></a>
<FONT color="green">110</FONT>            verifyBracketing(min, max);<a name="line.110"></a>
<FONT color="green">111</FONT>    <a name="line.111"></a>
<FONT color="green">112</FONT>            if (isBracketing(min, initial)) {<a name="line.112"></a>
<FONT color="green">113</FONT>                return solve(min, initial, fMin, fInitial);<a name="line.113"></a>
<FONT color="green">114</FONT>            } else {<a name="line.114"></a>
<FONT color="green">115</FONT>                return solve(initial, max, fInitial, fMax);<a name="line.115"></a>
<FONT color="green">116</FONT>            }<a name="line.116"></a>
<FONT color="green">117</FONT>        }<a name="line.117"></a>
<FONT color="green">118</FONT>    <a name="line.118"></a>
<FONT color="green">119</FONT>        /**<a name="line.119"></a>
<FONT color="green">120</FONT>         * Find a real root in the given interval.<a name="line.120"></a>
<FONT color="green">121</FONT>         *<a name="line.121"></a>
<FONT color="green">122</FONT>         * @param min Lower bound for the interval.<a name="line.122"></a>
<FONT color="green">123</FONT>         * @param max Upper bound for the interval.<a name="line.123"></a>
<FONT color="green">124</FONT>         * @param fMin function value at the lower bound.<a name="line.124"></a>
<FONT color="green">125</FONT>         * @param fMax function value at the upper bound.<a name="line.125"></a>
<FONT color="green">126</FONT>         * @return the point at which the function value is zero.<a name="line.126"></a>
<FONT color="green">127</FONT>         * @throws TooManyEvaluationsException if the allowed number of calls to<a name="line.127"></a>
<FONT color="green">128</FONT>         * the function to be solved has been exhausted.<a name="line.128"></a>
<FONT color="green">129</FONT>         */<a name="line.129"></a>
<FONT color="green">130</FONT>        private double solve(double min, double max,<a name="line.130"></a>
<FONT color="green">131</FONT>                             double fMin, double fMax)<a name="line.131"></a>
<FONT color="green">132</FONT>            throws TooManyEvaluationsException {<a name="line.132"></a>
<FONT color="green">133</FONT>            final double relativeAccuracy = getRelativeAccuracy();<a name="line.133"></a>
<FONT color="green">134</FONT>            final double absoluteAccuracy = getAbsoluteAccuracy();<a name="line.134"></a>
<FONT color="green">135</FONT>            final double functionValueAccuracy = getFunctionValueAccuracy();<a name="line.135"></a>
<FONT color="green">136</FONT>    <a name="line.136"></a>
<FONT color="green">137</FONT>            // [x0, x2] is the bracketing interval in each iteration<a name="line.137"></a>
<FONT color="green">138</FONT>            // x1 is the last approximation and an interpolation point in (x0, x2)<a name="line.138"></a>
<FONT color="green">139</FONT>            // x is the new root approximation and new x1 for next round<a name="line.139"></a>
<FONT color="green">140</FONT>            // d01, d12, d012 are divided differences<a name="line.140"></a>
<FONT color="green">141</FONT>    <a name="line.141"></a>
<FONT color="green">142</FONT>            double x0 = min;<a name="line.142"></a>
<FONT color="green">143</FONT>            double y0 = fMin;<a name="line.143"></a>
<FONT color="green">144</FONT>            double x2 = max;<a name="line.144"></a>
<FONT color="green">145</FONT>            double y2 = fMax;<a name="line.145"></a>
<FONT color="green">146</FONT>            double x1 = 0.5 * (x0 + x2);<a name="line.146"></a>
<FONT color="green">147</FONT>            double y1 = computeObjectiveValue(x1);<a name="line.147"></a>
<FONT color="green">148</FONT>    <a name="line.148"></a>
<FONT color="green">149</FONT>            double oldx = Double.POSITIVE_INFINITY;<a name="line.149"></a>
<FONT color="green">150</FONT>            while (true) {<a name="line.150"></a>
<FONT color="green">151</FONT>                // Muller's method employs quadratic interpolation through<a name="line.151"></a>
<FONT color="green">152</FONT>                // x0, x1, x2 and x is the zero of the interpolating parabola.<a name="line.152"></a>
<FONT color="green">153</FONT>                // Due to bracketing condition, this parabola must have two<a name="line.153"></a>
<FONT color="green">154</FONT>                // real roots and we choose one in [x0, x2] to be x.<a name="line.154"></a>
<FONT color="green">155</FONT>                final double d01 = (y1 - y0) / (x1 - x0);<a name="line.155"></a>
<FONT color="green">156</FONT>                final double d12 = (y2 - y1) / (x2 - x1);<a name="line.156"></a>
<FONT color="green">157</FONT>                final double d012 = (d12 - d01) / (x2 - x0);<a name="line.157"></a>
<FONT color="green">158</FONT>                final double c1 = d01 + (x1 - x0) * d012;<a name="line.158"></a>
<FONT color="green">159</FONT>                final double delta = c1 * c1 - 4 * y1 * d012;<a name="line.159"></a>
<FONT color="green">160</FONT>                final double xplus = x1 + (-2.0 * y1) / (c1 + FastMath.sqrt(delta));<a name="line.160"></a>
<FONT color="green">161</FONT>                final double xminus = x1 + (-2.0 * y1) / (c1 - FastMath.sqrt(delta));<a name="line.161"></a>
<FONT color="green">162</FONT>                // xplus and xminus are two roots of parabola and at least<a name="line.162"></a>
<FONT color="green">163</FONT>                // one of them should lie in (x0, x2)<a name="line.163"></a>
<FONT color="green">164</FONT>                final double x = isSequence(x0, xplus, x2) ? xplus : xminus;<a name="line.164"></a>
<FONT color="green">165</FONT>                final double y = computeObjectiveValue(x);<a name="line.165"></a>
<FONT color="green">166</FONT>    <a name="line.166"></a>
<FONT color="green">167</FONT>                // check for convergence<a name="line.167"></a>
<FONT color="green">168</FONT>                final double tolerance = FastMath.max(relativeAccuracy * FastMath.abs(x), absoluteAccuracy);<a name="line.168"></a>
<FONT color="green">169</FONT>                if (FastMath.abs(x - oldx) &lt;= tolerance ||<a name="line.169"></a>
<FONT color="green">170</FONT>                    FastMath.abs(y) &lt;= functionValueAccuracy) {<a name="line.170"></a>
<FONT color="green">171</FONT>                    return x;<a name="line.171"></a>
<FONT color="green">172</FONT>                }<a name="line.172"></a>
<FONT color="green">173</FONT>    <a name="line.173"></a>
<FONT color="green">174</FONT>                // Bisect if convergence is too slow. Bisection would waste<a name="line.174"></a>
<FONT color="green">175</FONT>                // our calculation of x, hopefully it won't happen often.<a name="line.175"></a>
<FONT color="green">176</FONT>                // the real number equality test x == x1 is intentional and<a name="line.176"></a>
<FONT color="green">177</FONT>                // completes the proximity tests above it<a name="line.177"></a>
<FONT color="green">178</FONT>                boolean bisect = (x &lt; x1 &amp;&amp; (x1 - x0) &gt; 0.95 * (x2 - x0)) ||<a name="line.178"></a>
<FONT color="green">179</FONT>                                 (x &gt; x1 &amp;&amp; (x2 - x1) &gt; 0.95 * (x2 - x0)) ||<a name="line.179"></a>
<FONT color="green">180</FONT>                                 (x == x1);<a name="line.180"></a>
<FONT color="green">181</FONT>                // prepare the new bracketing interval for next iteration<a name="line.181"></a>
<FONT color="green">182</FONT>                if (!bisect) {<a name="line.182"></a>
<FONT color="green">183</FONT>                    x0 = x &lt; x1 ? x0 : x1;<a name="line.183"></a>
<FONT color="green">184</FONT>                    y0 = x &lt; x1 ? y0 : y1;<a name="line.184"></a>
<FONT color="green">185</FONT>                    x2 = x &gt; x1 ? x2 : x1;<a name="line.185"></a>
<FONT color="green">186</FONT>                    y2 = x &gt; x1 ? y2 : y1;<a name="line.186"></a>
<FONT color="green">187</FONT>                    x1 = x; y1 = y;<a name="line.187"></a>
<FONT color="green">188</FONT>                    oldx = x;<a name="line.188"></a>
<FONT color="green">189</FONT>                } else {<a name="line.189"></a>
<FONT color="green">190</FONT>                    double xm = 0.5 * (x0 + x2);<a name="line.190"></a>
<FONT color="green">191</FONT>                    double ym = computeObjectiveValue(xm);<a name="line.191"></a>
<FONT color="green">192</FONT>                    if (FastMath.signum(y0) + FastMath.signum(ym) == 0.0) {<a name="line.192"></a>
<FONT color="green">193</FONT>                        x2 = xm; y2 = ym;<a name="line.193"></a>
<FONT color="green">194</FONT>                    } else {<a name="line.194"></a>
<FONT color="green">195</FONT>                        x0 = xm; y0 = ym;<a name="line.195"></a>
<FONT color="green">196</FONT>                    }<a name="line.196"></a>
<FONT color="green">197</FONT>                    x1 = 0.5 * (x0 + x2);<a name="line.197"></a>
<FONT color="green">198</FONT>                    y1 = computeObjectiveValue(x1);<a name="line.198"></a>
<FONT color="green">199</FONT>                    oldx = Double.POSITIVE_INFINITY;<a name="line.199"></a>
<FONT color="green">200</FONT>                }<a name="line.200"></a>
<FONT color="green">201</FONT>            }<a name="line.201"></a>
<FONT color="green">202</FONT>        }<a name="line.202"></a>
<FONT color="green">203</FONT>    }<a name="line.203"></a>




























































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